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1、概率论与数理统计英文题目Test 1. Consider the set W=1,2,3,4,5,6,7,8,9 with subsets A=1,3,5,7,9,Find the following sets: (a) AID and B=2,4,6,8,C=1,2,3,4,D=7,8 AUB (b) (CUD)c and AI(BUD) (EUF)c=EcIFc,(b)(EIF)c=EcUFc 2. ShowDe Morgans Law: (a) 3.(a)Let P(A)=0.5,P(B)=0.4,P( AB)=0.2. Find P(AB) and P(A|B) (b)Consider
2、 two fair dice A and B. Die A is six-sided and is numbered 1 through to 6 whilst die B is four-sided and is numbered 1 through 4. Both dice are rolled. Find the probability of two dice show the same score. 4.(a).Let X be a random variable. XN(m,s2). Show that Y=X-mN(0,1) s(b)The pmf of a random vari
3、able X which has a Poisson distribution with parameter 2. Find P(X=3). 5. Suppose (X,Y) be bivariate normal distribution. (X,Y)N(m1,s12,m2,s22;r). Find the correlation coefficient of X and Y. 6. (a)Let X Uniformly distributed on (-1,1), i.e. XU(-1,1). Find the Expectation and Variance of X. (b)Let X
4、 and Y be continuous random variables with joint pdf fX,Y(x,y)(x,y)R2. Let Z=X+Y, FindfX+Y(z). In particular, if X and Y are independent, Find fX*fY. 7.(a)The random variable X has pmf is: X 0 1 2 3 PX(x) 0.2 0.16 0.41 0.23 Find P(1X1) andP(X2X1). 8. X is continuous r.v. with pdf x12ef(x)=2x1e-22x0x
5、0Let Y=X, Find E(Y) 1 9.Let X be a continuous random variable with probability density function xe-xx0f(x)=otherwise0(a) Find P(1X0,y0 f(x)=otherwise0(a) Find the marginal pdf of X and Y. (b)Determine whether the two random variables are dependent or independent. 11Suppose the random variable X has
6、the density function 32x0f(x)=(x+4)30otherwise Find the probability density function of the random variable Y=X+4. 12. Take out a number from 1-200 ( 200 natural numbers). Find (a) the probability that this number can be drived by 6; (b) the probability that this number can be drived by 6 as well as
7、 8. 13. Three children are selected at random from a group of five boys and three girls. (a)What is the probability that all three are boys? (b)What is the probability at least two girls are selected? 14. (a)Let P(AB)=P(AB),P(A)p,findP(B). (b)LetP(A)=123，P(A|B)=，P(B|A)=.FindP(B). 53515.(a) Suppose A
8、, B are two events, and P(A)=1/4, P(B)=1/2, P(AB)=1/9, then evaluate PAB() (b)Use Bayes Theorem to show that if P(A),P(B)0 and p(A)p(A|B) then P(B)P(BA) 16.(a)Let X has a Binomial distribution with parameters n and p, i.e. X b (n, p).Find P(X=2) (b) Show that Cov(X,Y)=E(XY)-E(X)E(Y) 2 17. Suppose th
9、e distribution function of a random variable X is 0,x1Find (a) the probability that X gets value within (0.3，0.7); (b) the density function of X 18. The operational lifetime X, in years, of a battery powered watch has probability density function cx(6-x)3x6f(x)=otherwise 0 (a) Find the value of c. (
10、b) Find the cumulative distribution function of X. (c) Find the probability that the watch has an operational lifetime in excess of 4 years. 19.(a)The random variable X has the probability mass function below: X 1 2 3 4 0.14 Px(x) 0.4 a 0.24 Find a and FX(3.2) (b) A continuous random variable X havi
11、ng the probability density function x2f(x)=30 -1x2elsewhere.Find +-f(x)dx and P(0X1). 20.(a)Let P(BA)=P(BA),P(A)=1=P(B).FindP(AB) 3(b) Suppose X and Y are independent random variables XB(2,p),YB(3,p),and 5P(Y1)=.FindP(Y1). 921. Suppose the density function of (X,Y) is k(6-x-y),0x2,2y4f(x,y)= ,else0
12、(a) Determine the constant k. (b) Find the probability PX1,Y3. 22. Let X denote the number of times a certain numerical control machine will malfunction: 1,2,or 3 times on any given day. Let Y denote the number of times a technician is called on an emergency call. Their joint probability distributio
13、n is 3 given as Table 1.1 Table 1.1 X Y 1 2 3 1 0.05 0.05 0.1 2 0.05 0.1 0.35 3 0 0.2 0.1 (a) Evaluate the marginal distribution of X and Y (b) Determine whether the two random variables of X and Y are dependent or independent. 23.(a) IfXN(-2，0.42), then findE(X+3)2 (b) Let (X,Y) be 2-dimensional ra
14、ndom variables,and (X,Y)(1,0)(1,1)(2,0)(2,1)P0.40.2ab If E(XY)=0.8，find Cov(X,Y). 24.Suppose the probability distribution of Xi is as follows,andP(X1X2=0)=1. Find the correlation coefficient of X and Y. 25. Let X has a be uniform distribution on the interval (0,1),and Y=X2-4X+1 Find (a)fY(y) (b) E(Y
15、) 26. The random variable X, for fixed 0p0ef(x)=4elsewh.ere0, Find the mean and variance of the random variable Y. 4 29. Suppose the probability density function of random variable X is 1(x-1)1x9f(x)=32o.w.0, 1and Y=(X-1). 2Find (a) the probability density function of Y,fY(y). (b) expectation E(Y) and variance Var(Y). 30. Suppose X and Y are two random variables. The joint probability density function is 4xyf(x,y)=00x1,0y1otherwiseFind (a) the marginal density of X. (b) expectation E(Y) and variance Var(Y). (c)Determine whether the two random variables are dependent or independent ？ 5
